Special Session 155: Advances in mathematical modelling and numerical simulation of superfluids

Computational methods for the nonlinear Schroedinger equation with low regularity potential and nonlinearity

Weizhu Bao National University of Singapore Singapore Co-Author(s):

Abstract: In this talk, we begin with the nonlinear Schroedinger equation (NLSE) with different low regularity potentials and nonlinearities arising from modeling and simulation for quantum physics and chemistry, nonlinear/quantum optics, and quantum information and computation, etc. Optimal error bounds for time-splitting methods and exponential wave integrators are established for the NSLE under the proper regularity assumption on its solution determined by the low regularity potential and nonlinearity. Then we propose a novel symmetric and explicit Gautschi-type exponential wave integrator (sEWI) for the NLSE with low regularity potential and nonlinearity and establish its optimal error estimates under various regularity assumptions on potential and nonlinearity. Extensions to the NLSE with singular potentials and nonlinearities are presented. Finally, extensions to other dispersive PDEs with low regularity potential and nonlinearity are discussed. This talk is based on joint works with Remi Carles, Yue Feng, Bo Lin, Ying Ma, Chunmei Su, Qinglin Tang and Chushan Wang.

Supersolidity in Dipolar Bose-Einstein condensates

Blair P Blakie University of Otago New Zealand Co-Author(s):    Blair Blakie

Abstract: What happens when a solid starts flowing like a superfluid, yet still creaks like a crystal? This is the remarkable behaviour of a supersolid - a phase of matter that combines crystalline order with frictionless flow. Once a long-standing theoretical curiosity, supersolidity has now been realised in an exceptionally clean and tunable platform: dilute dipolar Bose-Einstein condensates of highly magnetic atoms such as dysprosium and erbium. Recent experiments have used the competing effects of short-range contact and long-range dipolar interactions to self-organise these ultracold gases into periodic density modulations, while maintaining global phase coherence. In this talk, I will explore the curious world of two-dimensional dipolar supersolids. These quantum materials break multiple symmetries and exhibit up to 3 different types of sound-wave excitations. Interestingly, this quantum fluid develops a shear-modulus marking solid-like rigidity, and exhibits transverse sound waves. I will discuss the underlying theory for the ground states and excitations, and how long-wavelength properties can be described by a hydrodynamic theory that incorporates both elasticity and superfluidity.

Dynamical reduction for solitonic filaments

Ricardo Carretero San Diego State University USA Co-Author(s):

Abstract: In this talk we describe techniques for the dynamical reduction of soliton stripes in nonlinear spatio-temporal systems. The central idea is to cast reductions to accurately describe these structures with lower-dimensional models that are more easily tackled, both mathematically and computationally. In turn, the reduced models allow for an unprecedented description of the statics, stability, dynamics, and interactions of these structures.

Numerical models for coupling Navier-Stokes and Gross-Pitaevskii solvers for two-fluid quantum flows

Ionut Danaila University of Rouen Normandy France Co-Author(s):

Abstract: In quantum flows, like liquid helium II at intermediate temperatures between zero and 2.17 K, a normal fluid and a superfluid coexist with independent velocity fields. The most advanced existing models for such systems use the Navier-Stokes equations for the normal fluid and a simplified description of the superfluid, based on the dynamics of quantized vortex filaments, with ad hoc reconnection rules. There was a single attempt [1] to couple Navier-Stokes and Gross-Pitaevskii equations in a global model intended to describe the compressible two-fluid liquid helium II. We present in this contribution a new numerical model to couple a Navier-Stokes incompressible fluid with a Gross-Pitaevskii superfluid [2]. A numerical algorithm based on pseudo-spectral Fourier methods is presented for solving the coupled system of equations. The new numerical system is validated against well-known benchmarks for the evolution in a normal fluid of different types or arrangements of quantized vortices (vortex crystal, vortex dipole and vortex rings). [1] C. Coste, Nonlinear Schrodinger equation and superfluid hydrodynamics, The European Physical Journal B - Condensed Matter and Complex Systems, VOL. 1, P. 245--253, 1998. [2] M. Brachet, G. Sadaka, Z. Zhang, V. Kalt and I. Danaila, Coupling Navier-Stokes and Gross-Pitaevskii equations for the numerical simulation of two-fluid quantum flows, J. Computational Physics, 488, p. 112193(1-17), 2023.

Scale-by-scale scalar statistics and extreme events in non-equilibrium multicomponent flows

Luminita Danaila University of Rouen Normandy France Co-Author(s):

Abstract: Scale-by-scale scalar statistics are investigated from first principles in non-equilibrium multicomponent flows, with particular emphasis on higher-order moments as signatures of rare and extreme events. The analysis focuses on how small-scale scalar fluctuations are shaped by large-scale gradients, advection, waves, coherent structures, and component coupling in systems where a single effective Reynolds number is insufficient to characterize the dynamics. The framework is illustrated primarily for quantum turbulence described by the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model, including temperature effects. A second case pertains to atmospheric flows simulated using the Weather Research and Forecasting (WRF) model in Large Eddy Simulation configuration, with application to heat-wave conditions over France. Although physically distinct, both systems exhibit strong mixing, variable thermodynamic properties, and a marked dependence of small-scale statistics on large-scale forcing. This parallel highlights common multiscale mechanisms governing intermittency and extreme events in non-equilibrium flows. 1. Polanco, J. I., Roche, P. E., Danaila, L., and Leveque, E. (2025). Disentangling temperature and Reynolds-number effects in quantum turbulence. Proceedings of the National Academy of Sciences, 122. 2. Zhang, Z., Danaila, L., Leveque, E., and Danaila, I. (2023). Higher-order statistics and intermittency in two-fluid quantum turbulence. Journal of Fluid Mechanics, 960, A6.

Numerical methods for Bogoliubov-de Gennes excitations of Bose-Einstein condensates

Yali Gao Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Yongyong Cai

Abstract: In this talk, we present the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs), which describe the elementary excitation of Bose-Einstein condensates around the mean-field ground state. Derived analytical properties of the BdGEs can serve as benchmark tests for numerical algorithms, and three numerical methods are proposed to solve the BdGEs, including the sine-spectral method, the central finite difference method, and the compact finite difference method. Extensive numerical tests are performed to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretisation, while the central finite difference method and the compact finite difference method have second and fourth order accuracy, respectively. Finally, the sine-spectral method is extended to study elementary excitations under the optical lattice potential and to solve the BdGEs around the first excited states of the GPE.

Superfluidity and phase transition in discrete clock symmetry

Michikazu Kobayashi Kochi University of Technology Japan Co-Author(s):

Abstract: Superfluidity in superfluid helium and ultracold atoms is characterized by broken continuous $U(1)$ symmetry. In contrast to these clean systems, superconductors are highly influenced by discrete symmetry originating from crystal structure and multi-band features. To investigate the influence of the discrete symmetry, we consider the two-dimensional Gross-Pitaevskii model having the discrete $\mathbb{Z}_n$ clock symmetry and the characteristics of the superfluid phase transition. The superfluid phase transition is characterized by the number $n$ and the strength $g$ of the discrete symmetry $\mathbb{Z}_n$. When $g$ is large, the superfluid phase transition is first order. When $g$ is small, on the other hand, the phase transition is 2nd order for $n \le 4$, and BKT-type for $n \ge 5$. Our numerical results are consistent with the variational renormalization group analysis.

Computing the action ground states of nonlinear Schr\odinger equation

Wei Liu National University of Defense Technology Peoples Rep of China Co-Author(s):

Abstract: This talk presents recent advances in numerical computation for the action ground state (GS) of the nonlinear Schr\odinger equation with possible rotational effects. We first establish computationally feasible variational characterizations for the action GS: minimization problems with the Nehari constraint or the $L^{p+1}$-normalization for both focusing and defocusing cases, and the unconstrained action minimization specifically for the defocusing case. Then, we develop effective numerical algorithms for computing the action GS by solving these variational problems, with rigorous convergence analysis. Furthermore, by combining theoretical analysis with numerical explorations, we present results regarding the properties of the action GS, in particular revealing its non-equivalence with the energy GS.

NLS equation with competing nonlinearities: orbital stability of kinks and solitons

Dmitry Pelinovsky McMaster University Canada Co-Author(s):

Abstract: Kinks connecting zero and nonzero equilibria in the NLS equation with competing nonlinearities occur at the special values of the frequency parameter. Since they are minimizers of energy, they are expected to be orbitally stable in the time evolution of the NLS equation. However, the stability proof is complicated by the degeneracy of kinks near the nonzero equilibrium. The main purpose of this work is to give a rigorous proof of the orbital stability of kinks. We give details of analysis and computations for the cubic-quintic NLS equation and show how the proof is extended to the general case. We also identify the orbitally stable solitons in the discrete version of this model and perform a comprehensive numerical study of all stable configurations in the anti-continuum limit of the model.

On the finite-time collapse problem for quantum vortices

Bartosz Protas McMaster University Canada Co-Author(s):    Vikas Krishnamurthy (IIT Hyderabad) and Dariche Nguyen (McMaster)

Abstract: In certain situations families of 2D point vortices may undergo a finite-time collapse where all vortices simultaneously converge at the same location such that their evolution cannot continue past that time. The best known scenario of this type concerns the self-similar collapse of three point vortices on the plane and was investigated by Gr{\o}bli already in 1877. We consider a quantum version of this problem where, in addition to their mutual induction, the vortices evolve under the influence of a trapping potential and argue that collapse may still occur in this setting. We propose a constructive approach based on asymptotic analysis where the collapsing configurations of quantum vortices are approximated in the limit of a weak trapping potential. It is based on a hierarchy of perturbation equations simplified using the self-similar structure of the original collapsing configuration. The collapsing configurations constructed in this manner are compared with their numerical approximations obtained by solving a multidimensional root-finding problem using a variant of Newton`s method.

Floquet geometric squeezing in fast-rotating condensates

Han Pu Rice University USA Co-Author(s):    Li Chen and Han Pu

Abstract: Constructing and manipulating quantum states in fast-rotating Bose-Einstein condensates (BECs) has long stood as a significant challenge as the rotating speed approached a critical velocity. Although a recent experiment [R. J. Fletcher {\it et al.}, Science {\bf 372}, 1318 (2021)] has realized the geometrically squeezed state of the guiding-center mode, the remaining degree of freedom, the cyclotron mode, remains unsqueezed due to the large energy gap of the Landau levels. To overcome this limitation, we propose a Floquet-based state-preparation protocol by periodically driving an anisotropic potential. Through both analytical and numerical calculations, we demonstrate that this protocol not only facilitates single-cyclotron-mode squeezing, but also enables two-mode squeezing. Such two-mode squeezing offers a richer set of dynamics compared to single-mode squeezing and can achieve a wave-packet width well below the lowest-Landau-level limit. Our work provides a highly controllable knob for realizing diverse geometrically squeezed states in ultracold quantum gases within the quantum Hall regime.

Coherent vortex equilibria in doubly periodic domains with background vorticity

Takashi Sakajo Kyoto University Japan Co-Author(s):

Abstract: We consider incompressible and non-viscous fluids in a two-dimensional domain with doubly periodic boundary condition, i.e. a flat torus. In this talk, I will present recent developments on the construction of stationary solutions consisting of coherent vortex structures such as point vortices and vortex patches. Unlike the case of unbounded plane, such a stationary solution on a bounded flat torus must satisfy the doubly periodic boundary condition and the Gauss constraint, where the vorticity in the domain must be zero. First, by using the theory of complex functions such as loxodromic functions and the explicit representation of Green`s function, we construct point vortex equilibria in which several Dirac`s delta singularities stay at the same locations. These stationary solutions have a constant background vorticity distribution other than point vortices. Next, through the regularizations of these point vortex equilibria, we obtain two classes of steady solutions to the incompressible Euler equations: one is the point vortex equilibria embedded in a continuous Liouville-type background vorticity distribution, and the other is the vortex patch equilibria with a background vorticity. This talk is based on the joint works with Vikas Krishnamurthy and Changjung Zou.

High-order parametric local discontinuous Galerkin methods for anisotropic curve-shortening flows

Chunmei Su Tsinghua University Peoples Rep of China Co-Author(s):    Xiuhui Guo, Wei Jiang, Chunmei Su

Abstract: We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For Pk approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal (k + 1)-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.